The free energy \({\cal F}={\cal F}_0+{\cal F}_{\rm int}\)  is usually decomposed into the sum of a noninteracting term \({\cal F}_0\)  and an interaction contribution \({\cal F}_{\rm int}\) .
To evaluate the interaction contribution to the Helmholtz free energy
we follow a familiar strategy [1]} by combining
a coupling constant integration expression for \({\cal F}_{\rm int}\)  valid for uniform continuum models (\(\hbar =1\)  from now on),
\({\cal F}_{\rm int}(T)=\frac{N}{2}\int _{0}^{1}d\lambda \int \frac{d^2{q}}{(2 \pi )^2}v_q\left[S^{(\lambda )}(q, T) - 1 \right]~,\) 
